Question: You have found the following ages (in years) of 6 sloths. Those sloths were randomly selected from the 41 sloths at your local zoo: $ 3,\enspace 1,\enspace 13,\enspace 1,\enspace 21,\enspace 14$ Based on your sample, what is the average age of the sloths? What is the standard deviation? You may round your answers to the nearest tenth.
Explanation: Because we only have data for a small sample of the 41 sloths, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\overline{x}} = \dfrac{3 + 1 + 13 + 1 + 21 + 14}{{6}} = {8.8\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {33.64} + {60.84} + {17.64} + {60.84} + {148.84} + {27.04}} {{6 - 1}} $ {s^2} = \dfrac{{348.84}}{{5}} = {69.77\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{69.77\text{ years}^2}} = {8.4\text{ years}} $ We can estimate that the average sloth at the zoo is 8.8 years old. There is also a standard deviation of 8.4 years.